A uniform wire of length / and radius r has a resistance of 100 `Omega`. It is recast into a wire of radius ` (r )/(2)`. The resistance of new wire will be:

Options

A. `1600 Omega`
B. `400 Omega`
C. `200 Omega`
D. `100 Omega`

Explanation

To solve the problem, we need to find the resistance of a new wire formed by recasting a uniform wire of length \( L \) and radius \( r \) with a known resistance of \( 100 \, \Omega \) into a wire of radius \( \frac{r}{2} \).

### Step-by-Step Solution:

1. **Understand the relationship between resistance, length, and area**:
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity of the material, \( L \) is the length of the wire, and \( A \) is the cross-sectional area.

2. **Calculate the area of the original wire**:
The cross-sectional area \( A \) of the original wire with radius \( r \) is:
\[ A = \pi r^2 \]

3. **Calculate the volume of the original wire**:
The volume \( V \) of the original wire can be calculated as:
\[ V = A \cdot L = \pi r^2 L \]

4. **Determine the area of the new wire**:
The new wire has a radius of \( \frac{r}{2} \). The cross-sectional area \( A' \) of the new wire is:
\[ A' = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} = \frac{\pi r^2}{4} \]

5. **Use the volume conservation principle**:
Since the volume remains constant during recasting, we have:
\[ V = A \cdot L = A' \cdot L' \]
Therefore,
\[ \pi r^2 L = \frac{\pi r^2}{4} \cdot L' \]
Simplifying this gives:
\[ L' = 4L \]

6. **Calculate the resistance of the new wire**:
Now we can find the resistance \( R' \) of the new wire using the resistance formula:
\[ R' = \frac{\rho L'}{A'} \]
Substituting \( L' \) and \( A' \):
\[ R' = \frac{\rho (4L)}{\frac{\pi r^2}{4}} = \frac{16 \rho L}{\pi r^2} \]

7. **Relate the new resistance to the original resistance**:
Since the original resistance \( R \) is given as \( 100 \, \Omega \):
\[ R = \frac{\rho L}{\pi r^2} = 100 \, \Omega \]
Therefore, we can express \( R' \) in terms of \( R \):
\[ R' = 16 R = 16 \times 100 \, \Omega = 1600 \, \Omega \]

### Final Result:
The resistance of the new wire will be:
\[ R' = 1600 \, \Omega \]